In this chapter, we will develop certain techniques that help solve problems stated in words. These techniques involve rewriting problems in the khung of symbols. For example, the stated problem
"Find a number which, when added to lớn 3, yields 7"
may be written as:
3 + ? = 7, 3 + n = 7, 3 + x = 1
and so on, where the symbols ?, n, và x represent the number we want lớn find. We gọi such shorthand versions of stated problems equations, or symbolic sentences. Equations such as x + 3 = 7 are first-degree equations, since the variable has an exponent of 1. The terms to lớn the left of an equals sign make up the left-hand member of the equation; those to the right hóa trang the right-hand member. Thus, in the equation x + 3 = 7, the left-hand thành viên is x + 3 và the right-hand thành viên is 7.
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SOLVING EQUATIONS
Equations may be true or false, just as word sentences may be true or false. The equation:
3 + x = 7
will be false if any number except 4 is substituted for the variable. The value of the variable for which the equation is true (4 in this example) is called the solution of the equation. We can determine whether or not a given number is a solution of a given equation by substituting the number in place of the variable and determining the truth or falsity of the result.
Example 1 Determine if the value 3 is a solution of the equation
4x - 2 = 3x + 1
Solution We substitute the value 3 for x in the equation and see if the left-hand thành viên equals the right-hand member.
4(3) - 2 = 3(3) + 1
12 - 2 = 9 + 1
10 = 10
Ans. 3 is a solution.
The first-degree equations that we consider in this chapter have at most one solution. The solutions to lớn many such equations can be determined by inspection.
Example 2 Find the solution of each equation by inspection.
a.x + 5 = 12b. 4 · x = -20
Solutions a. 7 is the solution since 7 + 5 = 12.b.-5 is the solution since 4(-5) = -20.
SOLVING EQUATIONS USING ADDITION and SUBTRACTION PROPERTIES
In Section 3.1 we solved some simple first-degree equations by inspection. However, the solutions of most equations are not immediately evident by inspection. Hence, we need some mathematical "tools" for solving equations.
EQUIVALENT EQUATIONS
Equivalent equations are equations that have identical solutions. Thus,
3x + 3 = x + 13, 3x = x + 10, 2x = 10, và x = 5
are equivalent equations, because 5 is the only solution of each of them. Notice in the equation 3x + 3 = x + 13, the solution 5 is not evident by inspection but in the equation x = 5, the solution 5 is evident by inspection. In solving any equation, we transform a given equation whose solution may not be obvious lớn an equivalent equation whose solution is easily noted.
The following property, sometimes called the addition-subtraction property, is one way that we can generate equivalent equations.
If the same quantity is added khổng lồ or subtracted from both membersof an equation, the resulting equation is equivalent khổng lồ the originalequation.
In symbols,
a - b, a + c = b + c, and a - c = b - c
are equivalent equations.
Example 1 Write an equation equivalent to
x + 3 = 7
by subtracting 3 from each member.
Solution Subtracting 3 from each member yields
x + 3 - 3 = 7 - 3
or
x = 4
Notice that x + 3 = 7 & x = 4 are equivalent equations since the solution is the same for both, namely 4. The next example shows how we can generate equivalent equations by first simplifying one or both members of an equation.
Example 2 Write an equation equivalent to
4x- 2-3x = 4 + 6
by combining like terms & then by adding 2 to each member.
Combining lượt thích terms yields
x - 2 = 10
Adding 2 lớn each thành viên yields
x-2+2 =10+2
x = 12
To solve an equation, we use the addition-subtraction property lớn transform a given equation khổng lồ an equivalent equation of the khung x = a, from which we can find the solution by inspection.
Example 3 Solve 2x + 1 = x - 2.
We want lớn obtain an equivalent equation in which all terms containing x are in one member and all terms not containing x are in the other. If we first showroom -1 khổng lồ (or subtract 1 from) each member, we get
2x + 1- 1 = x - 2- 1
2x = x - 3
If we now địa chỉ cửa hàng -x khổng lồ (or subtract x from) each member, we get
2x-x = x - 3 - x
x = -3
where the solution -3 is obvious.
The solution of the original equation is the number -3; however, the answer is often displayed in the size of the equation x = -3.
Since each equation obtained in the process is equivalent khổng lồ the original equation, -3 is also a solution of 2x + 1 = x - 2. In the above example, we can kiểm tra the solution by substituting - 3 for x in the original equation
2(-3) + 1 = (-3) - 2
-5 = -5
The symmetric property of equality is also helpful in the solution of equations. This property states
If a = b then b = a
This enables us lớn interchange the members of an equation whenever we please without having to be concerned with any changes of sign. Thus,
If 4 = x + 2thenx + 2 = 4
If x + 3 = 2x - 5then2x - 5 = x + 3
If d = rtthenrt = d
There may be several different ways lớn apply the addition property above. Sometimes one method is better than another, & in some cases, the symmetric property of equality is also helpful.
Example 4 Solve 2x = 3x - 9.(1)
Solution If we first add -3x to each member, we get
2x - 3x = 3x - 9 - 3x
-x = -9
where the variable has a negative coefficient. Although we can see by inspection that the solution is 9, because -(9) = -9, we can avoid the negative coefficient by adding -2x và +9 khổng lồ each thành viên of Equation (1). In this case, we get
2x-2x + 9 = 3x- 9-2x+ 9
9 = x
from which the solution 9 is obvious. If we wish, we can write the last equation as x = 9 by the symmetric property of equality.
SOLVING EQUATIONS USING THE DIVISION PROPERTY
Consider the equation
3x = 12
The solution khổng lồ this equation is 4. Also, cảnh báo that if we divide each thành viên of the equation by 3, we obtain the equations
whose solution is also 4. In general, we have the following property, which is sometimes called the division property.
If both members of an equation are divided by the same (nonzero)quantity, the resulting equation is equivalent to the original equation.
In symbols,
are equivalent equations.
Example 1 Write an equation equivalent to
-4x = 12
by dividing each thành viên by -4.
Solution Dividing both members by -4 yields
In solving equations, we use the above property khổng lồ produce equivalent equations in which the variable has a coefficient of 1.
Example 2 Solve 3y + 2y = 20.
We first combine like terms to lớn get
5y = 20
Then, dividing each thành viên by 5, we obtain
In the next example, we use the addition-subtraction property & the division property lớn solve an equation.
Example 3 Solve 4x + 7 = x - 2.
Solution First, we add -x and -7 lớn each member to get
4x + 7 - x - 7 = x - 2 - x - 1
Next, combining lượt thích terms yields
3x = -9
Last, we divide each thành viên by 3 to lớn obtain
SOLVING EQUATIONS USING THE MULTIPLICATION PROPERTY
Consider the equation
The solution lớn this equation is 12. Also, lưu ý that if we multiply each member of the equation by 4, we obtain the equations
whose solution is also 12. In general, we have the following property, which is sometimes called the multiplication property.
If both members of an equation are multiplied by the same nonzero quantity, the resulting equation Is equivalent lớn the original equation.
In symbols,
a = b và a·c = b·c (c ≠ 0)
are equivalent equations.
Example 1 Write an equivalent equation to
by multiplying each thành viên by 6.
Solution Multiplying each thành viên by 6 yields
In solving equations, we use the above property to lớn produce equivalent equations that are không lấy phí of fractions.
Example 2 Solve
Solution First, multiply each member by 5 to lớn get
Now, divide each thành viên by 3,
Example 3 Solve
Solution First, simplify above the fraction bar to lớn get
Next, multiply each thành viên by 3 khổng lồ obtain
Last, dividing each thành viên by 5 yields
FURTHER SOLUTIONS OF EQUATIONS
Now we know all the techniques needed to lớn solve most first-degree equations. There is no specific order in which the properties should be applied. Any one or more of the following steps listed on page 102 may be appropriate.
Steps to solve first-degree equations:Combine lượt thích terms in each thành viên of an equation.Using the addition or subtraction property, write the equation with all terms containing the unknown in one member and all terms not containing the unknown in the other.Combine lượt thích terms in each member.Use the multiplication property lớn remove fractions.Use the division property to obtain a coefficient of 1 for the variable.
Example 1 Solve 5x - 7 = 2x - 4x + 14.
Solution First, we combine like terms, 2x - 4x, khổng lồ yield
5x - 7 = -2x + 14
Next, we showroom +2x & +7 lớn each member and combine like terms khổng lồ get
5x - 7 + 2x + 7 = -2x + 14 + 2x + 1
7x = 21
Finally, we divide each thành viên by 7 khổng lồ obtain
In the next example, we simplify above the fraction bar before applying the properties that we have been studying.
Example 2 Solve
Solution First, we combine like terms, 4x - 2x, to lớn get
Then we địa chỉ cửa hàng -3 khổng lồ each thành viên and simplify
Next, we multiply each member by 3 to lớn obtain
Finally, we divide each member by 2 lớn get
SOLVING FORMULAS
Equations that involve variables for the measures of two or more physical quantities are called formulas. We can solve for any one of the variables in a formula if the values of the other variables are known. We substitute the known values in the formula and solve for the unknown variable by the methods we used in the preceding sections.
Example 1 In the formula d = rt, find t if d = 24 & r = 3.
Solution We can solve for t by substituting 24 for d và 3 for r. That is,
d = rt
(24) = (3)t
8 = t
It is often necessary lớn solve formulas or equations in which there is more than one variable for one of the variables in terms of the others. We use the same methods demonstrated in the preceding sections.
Example 2 In the formula d = rt, solve for t in terms of r và d.
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Solution We may solve for t in terms of r và d by dividing both members by r to lớn yield
from which, by the symmetric law,
In the above example, we solved for t by applying the division property to generate an equivalent equation. Sometimes, it is necessary khổng lồ apply more than one such property.